\newproblem{lay:1_8_26}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.8.26}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  \begin{enumerate}[a.]
		\item Show that the line through the vectors $\mathbf{p}$ and $\mathbf{q}$ in $\mathbb{R}^n$ may be written in parametric form as
		      $\mathbf{x}=(1-t)\mathbf{p}+t\mathbf{q}$.
		\item The line segment from $\mathbf{p}$ to $\mathbf{q}$ is the set of points of the form $(1-t)\mathbf{p}+t\mathbf{q}$ with $0\leq t\leq 1$
		      (as shown in the figure below). Show that a linear transformation maps this line segment onto a line segment or onto a single point.
	\end{enumerate}
}{
  % Solution
	\begin{enumerate}[a.]
		\item It is obvious that the line $\mathbf{x}=(1-t)\mathbf{p}+t\mathbf{q}$ goes through $\mathbf{p}$ (substitute $t=0$) and by $\mathbf{q}$ (substitute $t=1$).
		      We need to show that the locus of all these points is a line. To do so we rewrite it as\\
					\begin{center}
						$\mathbf{x}=(1-t)\mathbf{p}+t\mathbf{q}=\mathbf{p}+t(\mathbf{q}-\mathbf{p})$
					\end{center}
					that is the parametric form of a line.
		\item If we transform the points in the segment $\mathbf{y}=T(\mathbf{x})$ we have\\
					\begin{center}
						$\begin{array}{rcll}
							\mathbf{y}&=&T(\mathbf{x})& \text{By definition of }\mathbf{x}\\
												&=&T(\mathbf{p}+t(\mathbf{q}-\mathbf{p})) & \text{By linearity of }T\\
												&=&T(\mathbf{p})+tT(\mathbf{q}-\mathbf{p}) & \text{By linearity of }T\\
												&=&T(\mathbf{p})+tT(\mathbf{q})-tT(\mathbf{p}) & \\
												&=&(1-t)T(\mathbf{p})+tT(\mathbf{q}) & \\
						\end{array}$
					\end{center}
					If $T(\mathbf{q}-\mathbf{p})\neq \mathbf{0}$, then $\mathbf{y}$ describes a line that goes through $T(\mathbf{p})$ in the direction of $T(\mathbf{q}-\mathbf{p})$.
					If $T(\mathbf{q}-\mathbf{p})=\mathbf{0}$, then $\mathbf{y}$ is a single point.
	\end{enumerate}
}
\useproblem{lay:1_8_26}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
